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High-Order Quadratures for the Solution ofScattering Problems in Two Dimensions
Ran Duan and Vladimir RokhlinTechnical Report YALEU/DCS/TR-1401
April 22, 2008
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Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18
We construct an iterative algorithm for the solution of forward scattering problemsin two dimensions. The scheme is based on the combination of high-order quadratureformulae, fast application of integral operators in Lippmann-Schwinger equations,and the stabilized biconjugate gradient method (BI-CGSTAB). While the FFT-basedfast application of integral operators and the BI-CGSTAB for the solution of linearsystems are fairly standard, a large part of this paper is devoted to constructing a classof high-order quadrature formulae applicable to a wide range of singular functions intwo and three dimensions; these are used to obtain rapidly convergent discretizationsof Lippmann-Schwinger equations. The performance of the algorithm is illustratedwith several numerical examples.
High-Order Quadratures for the Solution ofScattering Problems in Two Dimensions
Ran Duan and Vladimir RokhlinTechnical Report YALEU/DCS/TR-1401
April 22, 2008
The authors were supported in part by the Schlumberger Technology Corporation, ONRGrant N00014-07-1-0711, and AFOSR Grant FA9550-06-1-0239.Approved for public release: Distribution is unlimited.Keywords: Scattering, Helmholtz, Lippmann-Schwinger, High-Order, Quadratures, Singu-lar, Hankel
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1 Introduction
Forward scattering has been an active field of research in science, mathematics, and engi-neering over the past several decades (see e.g. [3], [4]). The most straightforward method forthe solution of a forward scattering problem is to discretize the underlying PDEs directly,replace the derivatives with finite differences, and solve numerically the resulting system oflinear algebraic equations. However, discretization of differential equations leads to matri-ces with high condition numbers, with the attendant loss of accuracy, deterioration in theperformance of iterative methods, etc. Another approach is to convert the underlying PDEsinto integral equations of the second kind (normally referred to as the Lippmann-Schwingerequation), discretize the latter via appropriate quadrature formulae, and deal numericallywith the resulting linear systems. This paper constructs a class of high-order quadratureformulae applicable to the Lippmann-Schwinger equation in two and three dimensions.
1.1 Statement of the Problem
The forward scattering problem is the problem of determining the scattered field given theparameters of the scattering structure and the incident field. In this section, we formulatethe two-dimensional forward scattering problem for the Helmholtz equation, and derive thecorresponding Lippmann-Schwinger equation.
The forward scattering problem we investigate arises from the time domain wave equation
∂2
∂2tψ(x, t) = c2(x) · 2ψ(x, t), (1)
where ψ(x, t) is the value of the scalar field at a point x at time t, and c(x) is the local speedof wave propagation at a point x. In order to solve (1), we assume that
ψ(x, t) = ψk(x) ei k c0 t, (2)
where k is a complex number with non-negative imaginary part, and c0 is the speed of wavepropagation outside of the scattering structure. Substituting (2) into (1), we obtain
(2 + k2)ψk(x) = k2 V (x)ψk(x), (3)
whereV (x) = 1 − (
c0c(x)
)2. (4)
Equation (3) is the well-known Helmholtz equation, and the operator (2 + k2) is knownas the Helmholtz operator. For any point x outside the scattering object, c(x) = c0; therefore,V (x) = 0 outside the scattering object. We represent the field ψk(x) at a point x as a sumof two parts: the incident field ψin
k (x) and the scattered field ψscatk (x), i.e.,
ψk(x) = ψink (x) + ψscat
k (x). (5)
The incident field satisfies the homogenous Helmholtz equation
(2 + k2)ψink (x) = 0, (6)
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in some open region in R2 containing the scatterer; the scattered field satisfies the Sommerfeld
radiation condition
lim|x|→∞
√
|x| (∂ ψscatk (x)
∂ |x| − i k ψscatk (x)) = 0. (7)
Combining equations (3), (5), and (6), we obtain the equation for the scattered field
(2 + k2)ψscatk (x) − k2 V (x)ψscat
k (x) = k2 V (x)ψink (x). (8)
In this paper, we view the equation (8) with ψscatk satisfying the Sommerfeld condition (7)
as the principal formulation of the forward scattering problem. A standard approach to thenumerical solution of (8) is to convert (8) into the well-known Lippmann-Schwinger equation,which is an integral equation of the second kind, as follows (see, for example, [5]).
Convolving (8) with a Green’s function for the equation
(2 + k2)Gk(x, y) = δ(x− y), (9)
we obtain
ψscatk (x) − k2
∫
D
Gk(x, y)V (y)ψscatk (y) dy = k2
∫
D
Gk(x, y)V (y)ψink (y) dy, (10)
which is an integral equation of the second kind; in (10) above, D denotes the region in spacewhere the scatterer is located. As is well-known, in two dimensions, the Green’s functionGk(x, y) satisfying the condition (7) is
Gk(x, y) = − i
4H
(1)0 (k‖x− y‖), (11)
where H(1)0 (k‖x − y‖) is the Hankel function of the first kind of order zero. We will define
the operator L : L2(D) → L2(D) by the formula
L(ψ)(x) =
∫
D
Gk(x, y)V (y)ψ(y) dy, (12)
and observe that a large part of this paper is devoted to the construction of accurate dis-cretizations of L.
1.2 Overview
A number of algorithms exist for the modeling of acoustic scattering; since we are interestedin frequency domain results, we have concentrated on frequency domain (as opposed to time-domain) models. The usual approach to such problems is to convert the scattering probleminto the Lippmann-Schwinger equation, and solve the latter iteratively (integral equations ofthe second kind being much more amenable to iterative techniques than the straightforwarddiscretizations of underlying partial differential equations (PDEs)). In addition, the use ofthe Lippmann-Schwinger equation obviates the need to impose the radiation (Sommerfeld)condition at the boundary of the grid, since the “background” Green’s function (11) imposesthe Sommerfeld condition automatically.
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Historically, there have been two problems associated with the numerical use of integralequations in scattering calculations. First, the kernels of Lippmann-Schwinger equationsare dense, except when the background is extremely attenuating; since iterative techniquesrequire application of the matrix of the discretized integral operator to a sequence of re-cursively generated vectors, the cost of the procedure is prohibitive, except for extremelysmall-scale problems. This difficulty was overcome almost 40 years ago via the observa-tion that the free-space Green’s function for the Helmholtz equation is translation invariant;appropriately chosen discretizations of Lippmann-Schwinger equations result in Toeplitz ma-trices, and the latter can be rapidly applied to arbitrary vectors via the FFT (Fast FourierTransform), resulting in algorithms with CPU time requirements proportional to N · log(N),with N the number of nodes in the discretization of the problem. Various forms of thisapproach have been widely used in electrical engineering and other fields, under the name“k-space” methods; some of the existing codes are quite fast, even for discretizations involv-ing hundreds of millions of nodes. However, the resulting solvers for the underlying PDEsare usually not very accurate, due to the problem discussed in the following paragraph.
The second difficulty associated with numerical use of Lippmann-Schwinger equationsis due to the singular character of the Green’s function for the Helmholtz equation; intwo dimensions, the principal term of the singularity is of the form log(r), and in threedimensions, it is of the form 1/r. As a result, kernels of Lippmann-Schwinger equations aresingular; the singularities are located on the diagonal, and in two dimensions are of the form
K(x, y) = log(|x− y|) + P (x, y) · log(|x− y|) +Q(x, y), (13)
with P,Q two smooth functions, and P (x, x) = 0 for all x ∈ R2; the corresponding form inthree dimensions is
K(x, y) =1
|x− y| + P (x, y) · 1
|x− y| +Q(x, y). (14)
It is important to observe that in most cases, we do not have access to each of the functionsP,Q separately, but can only evaluate the whole kernel K given a pair of points (x, y). Inother words, standard integration techniques (such as product integration, etc.) can not beused efficiently. The standard procedure in the literature (referred to as the “singularityextraction”) is to subtract the principal singularity and treat it analytically, and apply thetrapezoidal quadrature rule to the remaining function. Since the latter is not smooth (havinginfinite derivatives at x = y), the procedure converges slowly, normally behaving like a secondorder scheme.
We introduce a class of quadrature formulae for functions of the form (13) in two di-mensions and (14) in three dimensions. The approach is somewhat related to the Ewaldsummation [6], and leads to quadratures that can be viewed as a version of the correctedtrapezoidal rule; it is easily combined with the FFT to obtain fast algorithms. While inprinciple corrections of arbitrarily high order could be constructed, in practice both thecomplexity of derivation and the number of corrections grow rapidly with the order. Wehave designed corrections of orders 4, 6, 8, and 10; they require 1, 5, 13, and 25 correctednodes respectively.
The paper is organized as follows. In Section 2, we summarize several well-known math-ematical facts to be used in the paper. In Section 3, we introduce analytical tools to be used
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in the construction of the algorithm. Section 4 describes the algorithm in detail, and containsa complexity analysis. In Section 5, several numerical examples are used to illustrate theperformance of the algorithm. Finally, Section 6 contains generalizations and conclusions.
2 Analytical Preliminaries
In this section, we summarize several well-known mathematical facts to be used in thesections below. All of these are either well known or easily derived from well-known results.
2.1 Notation
For an integer N ≥ 1, the two-dimensional discrete Fourier transform FN is a mappingconverting a two-dimensional complex sequence a = aj1j2, j1, j2 = −N, ..., N, into anothertwo-dimensional complex sequence A = Ak1k2, k1, k2 = −N, ..., N, defined by the formula
Ak1k2 =
N∑
j1=−N
N∑
j2=−N
aj1j2 e− 2πi
(2N+1)k1j1 e−
2πi
(2N+1)k2j2 . (15)
It is easily verified that the inverse (FN)−1 of the mapping FN is given by the formula
(FN)−1(A)j1j2 = aj1j2 =1
(2N + 1)2
N∑
k1=−N
N∑
k2=−N
Ak1k2 e2πi
(2N+1)k1j1 e
2πi
(2N+1)k2j2, (16)
with j1 = −N, ..., N , j2 = −N, ..., N .For a Helmholtz equation
∇2φ+ k2φ = 0 (17)
in two dimensions, the potential φ at a point x produced by a unit point source at x0 isgiven by the formula
φ(x) = − i
4H0(k‖x− x0‖), (18)
where k is a complex number such that Im(k) ≥ 0, and H0 is the Hankel function of the firstkind of order zero. The well-known Sommerfeld formula states that
H0(kr) =1
π·∫ ∞
−∞
1√k2 − λ2
· ei√
k2−λ2x · eiλy dλ (19)
for any k ∈ C+, r, x, y ≥ 0, and r =
√
x2 + y2 (see, for example, [9]).Finally, we will need the identity
n∑
j=0
fjgj = fn
n∑
k=0
gk −n−1∑
j=0
(fj+1 − fj) · (j
∑
k=0
gk), (20)
valid for two arbitrary finite sequences fj, j = 0, 1, 2, . . . , n,, gj, j = 0, 1, 2, . . . , n,. Byanalogy with integration by parts, (20) is normally referred to as summation by parts; it iseasily verified by a substitution.
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2.2 High-Order Corrected Trapezoidal Quadrature Rules for Sin-gular Functions in One Dimension
For a function f : [a, b] → R1 and integer n ≥ 2, the n-point trapezoidal rule Tn is definedby the formula
Tn(f) = h
( n−1∑
i=0
f(a+ ih) −(f(a) + f(b)
2
)
)
, (21)
where
h =b− a
n− 1, (22)
and is widely used as an approximation to the integral∫ b
af(x) dx. It is second order con-
vergent, as long as the second derivative of f is continuous on [a, b]. In other words, iff ∈ C2[a, b], then
∫ b
a
f(x) dx = Tn(f) +O(h2). (23)
For any function f ∈ C2m+2[a, b] with integer m ≥ 1, the well-known Euler-Maclaurinformula (see, for example, [1]) states that there exists a real number ξ with a < ξ < b, suchthat
∫ b
a
f(x) dx = Tn(f) +m
∑
l=1
h2lB2l
(2l)!(f (2l−1)(b) − f (2l−1)(a)) − h2m+2B2m+2
(2m+ 2)!f (2m+2)(ξ), (24)
where Bk, k = 0, 1, 2, ... denote the Bernoulli numbers (see [1]). It is easily seen from (24)that for any function f ∈ Cm[a−mh, b+mh] with integer m ≥ 3, it is possible to constructquadratures of the form
T nβm(f) = Tn(f) + h
m−12
∑
k=−m−12
(f(b+ kh) − f(a+ kh))βmk , (25)
where βmk are coefficients such that
∫ b
a
f(x) dx = T nβm(f) +O(hm+1), (26)
with h defined by (22). The quadrature T nβm is referred to as the (m+ 1)th-order endpoint-
corrected trapezoidal rule; for any given k and m, where m ≥ 3, and is odd, the coefficientβm
k can be obtained via a direct calculation (see [7]).While (21) and (25) are widely used for the numerical integration of smooth functions,
their use for singular integrands tends to encounter difficulties (see, for example, [11]). In[7], a class of quadrature formulae is constructed for approximating integrals of singularfunctions of the form
f(x) = φ(x) s(x), (27)
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where φ ∈ Cm[−b−mh, b+mh] with m ≥ 3, and s is an integrable function on [−b, b] with asingularity at zero. For integers n ≥ 1, p ≥ 1, and odd m ≥ 3, the quadrature Uτhβm for thefunctions with separable singularities (i.e., functions defined by (27)) is given by the formula
Uτhβm(f) = T ′nβm(f) +
p∑
j=−p
τhj φ(jh). (28)
In (28),
T ′nβm(f) = h
n∑
j=−nj 6=0
f(jh) −(
f(−b) + f(b)
2
)
+ h
m−12
∑
k=−m−12
(f(b+ kh) − f(−b+ kh))βmk ,
(29)where the coefficients βm
k can be obtained via direct calculation (see [7]), h = bn, and the
coefficients τhj can be obtained by solving the system of linear equations
p∑
j=−p
xi−1j τh
j =
∫ b
−b
(xi−1s(x)) dx− T ′nβm(xi−1s), (30)
with xj = jh, and i = 1, 2, . . . , 2p+ 1. The quadrature formula (28) is of order 2p+ 2.
Remark 2.1 While (21) is the standard trapezoidal rule, and (25) is the trapezoidal rulewith endpoints corrections, (28) is the trapezoidal rule with corrections at both endpointsand a singular point inside the interval. Here, the singular point is the center point, so thescheme is sometimes called a “center-corrected trapezoidal rule.” The coefficients τh
j arecalled correction coefficients.
Remark 2.2 The only difference between T ′nβm(f) and T n
βm(f) is that T ′nβm(f) does not
contain the term h · f(0), which may become infinite for f of the form (27).
2.3 Toeplitz Convolution
This section introduces two-dimensional Toeplitz convolutions and a procedure for the cal-culation of two-dimensional Toeplitz convolutions via the two-dimensional discrete Fouriertransform. The Toeplitz convolution a ∗ b of finite two-dimensional complex sequencesa = aj1j2, j1, j2 = −N, ..., N, and b = bj1j2, j1, j2 = −2N, ..., 2N, is defined by theformula
(a ∗ b)k1k2 =N
∑
j1=−N
N∑
j2=−N
aj1j2bk1−j1, k2−j2, (31)
where k1, k2 = −N, ..., N . The well-known convolution theorem states that the Toeplitzconvolution a ∗ b is equal to the inverse Fourier transform of the product of the Fouriertransform of a′ and b, where a′ is a two-dimensional sequence obtained by padding thetwo-dimensional sequence a with zeros. In other words,
(a ∗ b)k1k2 = (F2N)−1(
F2N(a′) · F2N(b))
k1k2, (32)
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where k1, k2 = −N, ..., N , and the coefficients of the two-dimensional complex sequencea′ = a′i1i2
, i1, i2 = −2N, ..., 2N are defined by the formulae
a′i1i2= ai1i2 , (33)
when −N ≤ i1, i2 ≤ N , otherwisea′i1i2
= 0. (34)
Remark 2.3 While direct calculation of Toeplitz convolution (31) leads to time cost of orderO(N4), which is prohibitive for large scale problems, application of FFT to the formula (32)reduces the time cost to O(N2 · logN) (see, for example, [2]). In this paper, FFT is used forthe fast calculation of Toeplitz convolution.
3 Mathematical Apparatus
In this section, we introduce analytical tools to be used in the construction of the algorithms.
3.1 Endpoint Corrected Trapezoidal Quadrature Rules in Two Di-mensions
This section can be viewed as the extension of results of Section 2.2 to two dimensions.For a function f : [a, b] × [a, b] → R1 and integer n ≥ 2, the two-dimensional trapezoidal
rule T 2Dn is defined by the formula
T 2Dn (f) =
n−1∑
i=0
n−1∑
j=0
f(a+ ih, a+ jh) · h2 · βij , (35)
where
h =b− a
n− 1, (36)
and βij equals 1 in the interior of the square [a, b]× [a, b], equals 12
in the interior of the edge,and equals 1
4on the corners of the square.
Further, if f ∈ C2[a, b] × [a, b], then
∫ b
a
∫ b
a
f(x, y) dx dy = T 2Dn (f) +O(h2). (37)
The proof consists of a straightforward application of one-dimensional trapezoidal rule (21)to both directions in two dimensions, and thus omitted.
The following Lemma provides the two-dimensional version of the Euler-Maclaurin for-mula.
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Lemma 3.1 Suppose that a function f ∈ C2m+2[a, b]× [a, b], integers m ≥ 1, n ≥ 2 . Then,
∫ b
a
∫ b
a
f(x, y) dx dy = T 2Dn (f) +
i=n−1∑
i=0
m∑
l=1
h2l+1 ·B2l · βi
(2l)!·
(
∂2l−1
∂x2l−1f(b, a + ih) − ∂2l−1
∂x2l−1f(a, a+ ih) − ∂2l−1
∂y2l−1f(a+ ih, b) − ∂2l−1
∂y2l−1f(a+ ih, a)
)
+m
∑
l=1
m∑
l′=1
h2l+2l′ ·B2l · B2l′
(2l!)(2l′!)·(
∂2l+2l′−2
∂x2l−1 ∂y2l′−1
(
f(b, b)+f(a, a)−f(a, b)−f(b, a))
)
+O(h2m+2),
(38)
where T 2Dn (f) is defined in (35), h is defined in (36),
βi =
1 0 < i < n− 11/2 i = 0 or i = n− 1
, (39)
and Bk, k = 0, 1, 2, ... denote the Bernoulli numbers.
Proof. The Euler-Maclaurin formula (24) can be rewritten as∫ b
a
f(x) dx = Tn(f) +m
∑
l=1
h2lB2l
(2l)!(f (2l−1)(b) − f (2l−1)(a)) +O(h2m+2). (40)
Hence,∫ b
a
∫ b
a
f(x, y) dx dy =
∫ b
a
(
n−1∑
i=0
f(a+ ih, y) · βi +
m∑
l=1
h2lB2l
(2l)!(f (2l−1)(b) − f (2l−1)(a)) +O(h2m+2)
)
dy. (41)
The conclusion of the Lemma follows immediately from applying the formula (40) to theintegrals in (41).
The following Lemma provides a (2m+ 2)th-order endpoint-corrected trapezoidal rule intwo dimensions.
Lemma 3.2 Suppose that a function f ∈ C2m+2[a, b] × [a, b] with integers m ≥ 1, n ≥ 2,h = b−a
n−1. Then,
∫ b
a
∫ b
a
f(x, y) dx dy = T 2D,n
β2m+1(f) +O(h2m+2), (42)
In (42),
T 2D,n
β2m+1(f) = T 2Dn (f) + h2
n−1∑
i=0
m∑
k=−m
(
β2m+1k βi·
(
f(b+ kh, ih)+ f(ih, b+ kh)− f(a+ kh, ih)− f(ih, a+ kh))
)
+h2m
∑
k=−m
m∑
k′=−m
(
β2m+1k β2m+1
k′ ·
(
f(a+ kh, a + k′h) + f(b+ kh, b+ k′h) − f(a+ kh, b+ k′h) − f(b+ kh, a+ k′h))
)
, (43)
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T 2Dn , β2m+1
k , and βi are defined by (35), (25) and (39).
Proof. It is easily observed from the formulae (24), (25) that
m∑
l=1
h2l+1B2l
(2l!)f (2l−1)(a) =
m∑
k=−m
f(a+ kh) · β2m+1k · h2. (44)
Therefore,
m∑
k=−m
h2 · f(a+ kh, a+ ih) · β2m+1k =
m∑
l=1
h2l+1B2l
(2l!)
∂2l+1
∂x2l+1f(a, a+ ih), (45)
and
m∑
l=1
m∑
l′=1
h2l+2l′ · B2l ·B2l′
(2l!)(2l′!)
(
∂2l+2l′−2
∂x2l−1 ∂y2l′−1f(a, a)
)
=
m∑
l′=1
h2l′B2l′
(2l′!)·(
m∑
k=−m
h · β2m+1k
∂2l′−1
∂y2l′−1f(a+ kh, a)
)
=m
∑
k=−m
h2β2m+1k ·
(
m∑
l′=1
h2l′−1B2l′
(2l′!)· ∂
2l′−1
∂y2l′−1f(a+ kh, a)
)
=
m∑
k=−m
m∑
k′=−m
h2 · β2m+1k β2m+1
k′ f(a+ kh, a+ k′h). (46)
Now, (43) follows immediately from the combination of (45), (46).
3.2 High-Order Center Corrected Trapezoidal Quadrature Rulesfor the Singular Functions in Two Dimensions
The following Lemma provides an estimate of the difference between the integral and theend-point corrected trapezoidal quadrature for functions of the form x2p+2 · log(x2 + y2).
Lemma 3.3 Suppose that n is a positive integer, a, h are two positive real numbers suchthat h = a/n, and integers m, p are such that m ≥ p+ 1 ≥ 1. Then,
∫ a
−a
∫ a
−a
(x2p+2 · log(x2 + y2)) dx dy = T 2D,n
β2m+1(x2p+2 · log(x2 + y2)) +O(h2p+4), (47)
with T 2D,n
β2m+1 defined by (43).
Proof. Here, we prove the case of p = 0 as an illustration; the proof for p > 0 is quitesimilar. For simplicity, we will be denoting x2 · log(x2 + y2) by g(x, y). Then,
∂g(x, y)
∂y=
2x2y
x2 + y2, (48)
11
∂3g(x, y)
∂y3=
16x2y3
(x2 + y2)3 − 12x2y
(x2 + y2)2 , (49)
and∂6g(x, y)
∂y6= − 7680x2y6
(x2 + y2)6 +11520x2y4
(x2 + y2)5 − 4320x2y2
(x2 + y2)4 +240x2
(x2 + y2)3 . (50)
Replacing the derivatives in (24) with the appropriate finite differences, we rewrite (24) inthe form
∫ b
a
g(x) dx = Tn(g) +1
12h · (−g(b+ h) + g(a+ h) + g(b− h) − g(a− h))+
C1 · h4 · (g(3)(b) − g(3)(a)) − C2 · h6 · g(6)(ξ)
= T nβ3(g) + C1 · h4 · (g(3)(b) − g(3)(a)) + C2 · h6 · g(6)(ξ), (51)
where C1 and C2 are two constants independent of g, and ξ ∈ [a, b]. Therefore, for anyfixed x and g(x, y) = x2 · log(x2 + y2), the error ǫ(x) of the end-point corrected trapezoidalquadrature in the y direction is
ǫ(x) =
∫ a
−a
g(x, y) dy − T nβ3(g) = C1 · h4
(
16x2a3
(x2 + a2)3− 12x2a
(x2 + a2)2
)
+ C2 · h6 · g(6)(x, ξ). (52)
Summing up all the errors along the x axis, we obtain
ǫ =
∫ a
−a
∫ a
−a
(x2 · log(x2 + y2))dxdy − T 2D,n
β3 (x2 · log(x2 + y2)) ∼ 2 ·n
∑
i=1
h · ǫ(ih)
≤ h ·n
∑
i=1
(
C1 · h4
(
16(ih)2a3
((ih)2 + a2)3− 12(ih)2a
((ih)2 + a2)2
)
+ h ·n
∑
i=1
C2 · h6 · |g(6)(ξ)|max
≤ C1 h4
∫ a
0
(
16x2a3
(x2 + a2)3− 12x2a
(x2 + a2)2
)
dx+ C2 · h7
n∑
i=1
240
(ih)4
∼ C1 · (3 − π) · h4 + C2 · h6 · log(h) ∼ O(h4). (53)
Theorem 3.4 below is an extension of the formula (28) to two dimensions. For an integerp ≥ 0, it supplies a (2p + 4)th-order center-corrected quadrature formula on R
2 for thefunctions of the form
f(x, y) = φ(x, y) · s(x, y), (54)
where φ : R2 → R, and
s(x, y) = log(x2 + y2) + P (x, y) · log(x2 + y2) +Q(x, y), (55)
with P,Q two smooth functions, and P (0, 0) = 0. Suppose that n, m are positive integers,and a, h are two positive real numbers such that h = a/n. We define T ′2D,n
β2m+1(f) by the
12
formula
T ′2D,n
β2m+1(f) =∑
(i,j)∈M
f(ih, jh) · h2 · βij + h2n
∑
i=−n
m∑
k=−m
(
β2m+1k βi·
(f(a+ kh, ih) + f(ih, a+ kh) − f(−a + kh, ih) − f(ih,−a+ kh)))
+h2
m∑
k=−m
m∑
k′=−m
(
β2m+1k β2m+1
k′ ·
(f(−a+ kh,−a + k′h) + f(a+ kh, a + k′h) − f(−a+ kh, a + k′h) − f(a+ kh,−a + k′h)))
,
(56)
whereM = i, j ∈ Z : |i| ≤ n, |j| ≤ n, (i, j) 6= (0, 0), (57)
βi =
1 |i| < n1/2 |i| = n
; (58)
in (56), coefficients βij equal 1 in the interior of the square [−a, a] × [−a, a], equal 12
in theinterior of the edge, and equal 1
4on the corners of the square, and β2m+1
k are defined in (25).
Theorem 3.4 Suppose that φ ∈ C2p+2(R2) with p ≥ 0, integer m ≥ p+1, and s is a singularfunction on R2 with a logarithmic singularity at (0, 0), i.e., of the form (55). Suppose furtherthat
U2Dτhβ2m+1(φ · s) = T ′2D,n
β2m+1(φ · s) +∑
(i,j)∈W
τhij φ(ih, jh), (59)
where T ′2D,n
β2m+1(φ · s) is defined by the formula (56),
W = i, j ∈ Z : |i+ j| ≤ p and |i− j| ≤ p, (60)
and the coefficients τhij in (59) satisfy the system of linear equations
∑
(i,j)∈W
xi′−1i yj′−1
j τhij =
∫ a
−a
∫ a
−a
(xi′−1yj′−1s(x, y)) dx dy − T ′2D,n
β2m+1(xi′−1yj′−1s), (61)
with xi = ih, yj = jh, and (i′, j′) ∈ H, where H = i′, j′,∈ Z : i′ ≥ 1, j′ ≥ 1, i′+j′ ≤ 2p+2.Then,
∫ a
−a
∫ a
−a
(φ(x, y) · s(x, y)) dx dy = U2Dτhβ2m+1(φ · s) +O(h2p+4). (62)
Proof. Applying the Taylor expansion to the function φ(x, y) at the point (0, 0) we have
φ(x, y) = P (x, y) +R(x, y), (63)
where
P (x, y) =
2p+1∑
j=0
j∑
i=0
1
j!
(
j
i
)
xiyj−i ∂j
∂xi∂yj−iφ(x, y)|x=0, y=0 , (64)
13
and
R(x, y) =1
(2p+ 2)!
2p+2∑
i=0
(
2p+ 2
i
)
xiy2p+2−i ∂2p+2
∂xi∂y2p+2−iφ(x, y)|x=ξ1, y=ξ2 , (65)
where ξ1, ξ2 ∈ [−a, a] × [−a, a]. Thus,
|∫ a
−a
∫ a
−a
(φ(x, y) · s(x, y)) dx dy − U2Dτhβ2m+1(φ · s)|
6 |∫ a
−a
∫ a
−a
(P (x, y) · s(x, y)) dx dy − U2Dτhβ2m+1(P · s)|+
|∫ a
−a
∫ a
−a
(R(x, y) · s(x, y)) dx dy − U2Dτhβ2m+1(R · s)|. (66)
Now, we make the following three observations. Due to (61),
∫ a
−a
∫ a
−a
(P (x, y) · s(x, y)) dx dy − U2Dτhβ2m+1(P · s) = 0, (67)
and due to (59),
|∫ a
−a
∫ a
−a
(R(x, y) · s(x, y)) dx dy − U2Dτhβ2m+1(R · s)|
6 |∫ a
−a
∫ a
−a
(R(x, y) · s(x, y)) dx dy − T ′2D,n
β2m+1(R · s)| + |∑
(i,j)∈W
τhij R(ih, jh)|. (68)
Due to formulae (47) and (61), it is obvious that
|∫ a
−a
∫ a
−a
(R(x, y) · s(x, y)) dx dy − T ′2D,n
β2m+1(R · s)| ∼ O(h2p+4),
(69)
and the coefficients τhij are of the order h2, hence,
|∑
(i,j)∈W
τhij R(ih, jh)| ∼ O(h2p+4). (70)
Finally, the conclusion of the Lemma follows from the combination of (66) – (70).
3.3 High-Order Center Corrected Trapezoidal Quadrature Rulesfor the Green’s Function for the Helmholtz equation
In this section, we prove Theorem 3.10, which is the principal analytical tool of this paper.Theorem 3.10 describes the 10th-order center-corrected trapezoidal quadrature formulae forthe Hankel function. It can be viewed as a special case of Theorem 3.4 with p = 3, and thesingular function s the Hankel function H0.
14
In the remainder of this paper, we will be using the following notation. For any k ∈ C+
and h > 0, we will define the complex numbers D0, D1, D2, D3, D4, D5, via the formulae
D0 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr) dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · h2, (71)
D1 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr)x2 dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · (ph)2 · h2, (72)
D2 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr)x4 dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · (ph)4 · h2, (73)
D3 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr)x2y2 dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · (ph)2(qh)2 · h2, (74)
D4 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr)x6 dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · (ph)6 · h2, (75)
D5 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr)x4y2 dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · (ph)4(qh)2 · h2. (76)
The following Lemma is a simple consequence of the Sommerfeld formula (19).
Lemma 3.5 For any k ∈ C+, r, x, y ≥ 0, and r =
√
x2 + y2,
H0(kr) =1
π·∫ ∞
−∞
1√k2 − λ2
· ei·√
22·(√
k2−λ2−λ)·x · ei·√
22·(√
k2−λ2+λ)·y dλ, (77)
where r2 = x2 + y2, x, y ≥ 0.
The following two technical lemmas follow immediately from the Sommerfeld formula (19).
Lemma 3.6 For any k ∈ C+, and a ≥ 0,
∫ a
−a
∫ a
−a
H0(kr) dx dy =4
π·∫ ∞
−∞
1√k2 − λ2
·
ei·√
22·(√
k2−λ2−λ)·a − 1
i ·√
22· (√k2 − λ2 − λ)
· ei·√
22·(√
k2−λ2+λ)·a − 1
i ·√
22· (√k2 − λ2 + λ)
dλ, (78)
with r =√
x2 + y2.
Proof. Substituting (77) into the left side of (78), and changing the order of integration,we obtain
∫ a
−a
∫ a
−a
H0(kr) dx dy = 4
∫ a
0
∫ a
0
H0(kr) dx dy
=4
π
∫ ∞
−∞
dλ√k2 − λ2
·∫ a
0
ei·√
22·(√
k2−λ2−λ)·x dx ·∫ a
0
ei·√
22·(√
k2−λ2+λ)·y dy
=4
π·∫ ∞
−∞
1√k2 − λ2
· ei·√
22·(√
k2−λ2−λ)·a − 1
i ·√
22· (√k2 − λ2 − λ)
· ei·√
22·(√
k2−λ2+λ)·a − 1
i ·√
22· (√k2 − λ2 + λ)
dλ. (79)
15
Lemma 3.7 For any k ∈ C+, integer n ≥ 1, and a > 0,
n∑
p=−n
n∑
q=−n
H0(k√
(ph)2 + (qh)2) · h2 · βpq (80)
=4
π
∫ ∞
−∞
1√k2 − λ2
· h2 ·(
ei·√
22·(√
k2−λ2−λ)·a − 1
ei·√
22·(√
k2−λ2−λ)·h − 1− 1
2+
1
2ei·
√2
2·(√
k2−λ2−λ)·a)
·(
ei·√
22·(√
k2−λ2+λ)·a − 1
ei·√
22·(√
k2−λ2+λ)·h − 1− 1
2+
1
2ei·
√2
2·(√
k2−λ2+λ)·a)
dλ, (81)
withh = a/n, (82)
and βpq equals 1 in the interior of the (2n+ 1)× (2n+ 1) square, equals 12
in the interior ofthe edge, and equals 1
4on the corners of the square.
Proof. The trapezoidal sum (84) over the domain [−a, a] × [−a, a] is equal to four timesthe trapezoidal sum over the domain [0, a] × [0, a]. In other words,
n∑
p=−n
n∑
q=−n
H(1)0 (k
√
(ph)2 + (qh)2) · h2 · βpq
= 4 ·n
∑
p=0
n∑
q=0
H(1)0 (k
√
(ph)2 + (qh)2) · h2 · β ′pq, (83)
where β ′pq equals 1 in the interior of the (n+1)×(n+1) square, equals 1
2in the interior of the
edge, and equals 14
on the corners of the square. Substituting (77) into (83), and exchangingthe order of integration and summation, we obtain
n∑
p=−n
n∑
q=−n
H0(k√
(ph)2 + (qh)2) · h2 · βpq (84)
4
π
∫ ∞
−∞
dλ√k2 − λ2
· h2 ·( n
∑
p=0
ei·√
22·(√
k2−λ2−λ)·p·h − 1 + ei·√
22·(√
k2−λ2−λ)·a
2
)
·( n
∑
q=0
ei·√
22·(√
k2−λ2+λ)·q·h − 1 + ei·√
22·(√
k2−λ2+λ)·a
2
)
(85)
=4
π
∫ ∞
−∞
1√k2 − λ2
· h2 ·(
ei·√
22·(√
k2−λ2−λ)·a − 1
ei·√
22·(√
k2−λ2−λ)·h − 1− 1
2+
1
2ei·
√2
2·(√
k2−λ2−λ)·a)
·(
ei·√
22·(√
k2−λ2+λ)·a − 1
ei·√
22·(√
k2−λ2+λ)·h − 1− 1
2+
1
2ei·
√2
2·(√
k2−λ2+λ)·a)
dλ. (86)
.
16
Remark 3.1 As a → ∞, the exponential terms e(i·√
22·(√
k2−λ2±λ)·a) in (78) and (81) tend tozero; this fact will be used in Lemma 3.8 below.
The following lemma provides an analytical form for the difference between the inte-gral (78) and the trapezoidal sum (81). Its proof consists of combining Remark 3.1 with(78), (81).
Lemma 3.8 For any k ∈ C+ and h > 0,
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr) dx dy −
∞∑
p=−∞
∞∑
q=−∞H
(1)0 (k
√
(ph)2 + (qh)2) · h2
=4
π
∫ ∞
−∞
dλ√k2 − λ2
·(
1
(iα1)(iα2)− h2
4
eiα1h + 1
eiα1h − 1
eiα2h + 1
eiα2h − 1
)
, (87)
wherer =
√
x2 + y2, (88)
α1 =
√2
2(√k2 − λ2 − λ), α2 =
√2
2(√k2 − λ2 + λ). (89)
The following Lemma follows immediately from Lemma 3.8. It supplies an analyticalform for the difference∫ ∞
−∞
∫ ∞
−∞
(
xi−1yj−1H0(k√
x2 + y2))
dx dy−∑
(p,q)6=(0,0)
(
(ph)i−1(qh)j−1H0(k√
(ph)2 + (qh)2))
·h2,
(90)with i = 1, j = 1.
Remark 3.2 (90) is the right hand side of the equation (61), and thus is directly used inthe calculation of coefficients τh
ij. Direct numerical subtraction of the integral and the sum in(90) leads to loss of accuracy because of cancellation errors, especially when i, j are relativelylarge. Lemma 3.9 below and Lemmas 6.1 – 6.5 in Appendix A, provide analytical formulaefor (90) with (i, j) = (1, 1), (3, 1), (5, 1), (3, 3), (7, 1), (5, 3), i.e., D0 – D5 defined by (71) –(76), so that cancellation errors are reduced.
Lemma 3.9 For any k ∈ C+ and h > 0,
D0 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr) dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · h2
=4
π
∫ ∞
−∞
dλ√k2 − λ2
·(
1
(iα1)(iα2)− h2
2· eiα1h + eiα2h
(eiα1h − 1)(eiα2h − 1)
)
(91)
=4
π
∫ ∞
−∞
dλ√k2 − λ2
·(λ2
4− k2
24)h2 + z1 + z2 + i
√
12λh(y1 − y2) + y1y2
(iα1)(iα2)(1 + x1)(1 + x2), (92)
where the complex numbers x1, x2, y1, y2, z1, z2 are defined by the formulae
x1 =eiα1h − 1
iα1h− 1 =
∞∑
n=1
(iα1h)n
(n+ 1)!, x2 =
eiα2h − 1
iα2h− 1 =
∞∑
n=1
(iα2h)n
(n + 1)!, (93)
17
y1 = x1 −iα1h
2=
∞∑
n=2
(iα1h)n
(n+ 1)!, y2 = x2 −
iα2h
2=
∞∑
n=2
(iα2h)n
(n+ 1)!, (94)
z1 = y1 −(iα1h)
2
6=
∞∑
n=3
(iα1h)n
(n+ 1)!, z2 = y2 −
(iα2h)2
6=
∞∑
n=3
(iα2h)n
(n+ 1)!, (95)
and r, α1, and α2 are defined by (88), (89).
Proof. Substituting (77) into∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · h2, (96)
and exchanging the order of integration and summation, we obtain∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · h2
=4
π
∫ ∞
−∞
dλ√k2 − λ2
·(
h2
4
eiα1h + 1
eiα1h − 1
eiα2h + 1
eiα2h − 1− h2
4
)
=4
π
∫ ∞
−∞
dλ√k2 − λ2
· h2
2· eiα1h + eiα2h
(eiα1h − 1)(eiα2h − 1). (97)
Now, (91) follows immediately from the combination of (97), (78), and Remark 3.1. Substi-tuting (93) into (91), we obtain
1
(iα1)(iα2)− h2
2· eiα1h + eiα2h
(eiα1h − 1)(eiα2h − 1)
=1
(iα1)(iα2)− 1
2· 2 + iα1h(1 + x1) + iα2h(1 + x2)
(iα1)(iα2)(1 + x1)(1 + x2)
=x1 + x2 + x1x2 − 1
2iα1h(1 + x1) − 1
2iα1h(1 + x1)
(iα1)(iα2)(1 + x1)(1 + x2). (98)
Finally, (92) follows from the combination of (93), (94), (95) and (98).
Remark 3.3 Introducing the notation z = λk, we rewrite D0 in the form
D0 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr) dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · h2
=4h2
π
∫ ∞
−∞
d z√1 − z2
·(
1
(z2 − 0.5) · (kh)2− 1
2· ei
√2
2(√
1−z2−z)kh + ei√
22
(√
1−z2+z)kh
(ei√
22
(√
1−z2−z)kh − 1)(ei√
22
(√
1−z2+z)kh − 1)
)
.
(99)
Thus, D0 is entirely determined by k and h, and is of the form h2 · f(k · h). Similarly, D1 isof the form h4 · f(k · h); D2 and D3 are of the form h6 · f(k · h); D4 and D5 are of the formh8 · f(k · h). In other words, except for the multiplicative factors (h2,h4, h6, or h8), D0 – D5
only depend on the product k · h.
18
Remark 3.4 Even when Lemmas 6.1 – 6.5 are used, a certain loss of accuracy in the cal-culation of D1 – D5 is encountered (see Remark 3.2 above). Thus, evaluating D0 in doubleprecision, one obtains roughly 13 digits; for D1 one gets 9 digits, and D2 D3 D4 D5 yieldeven fewer digits.
To avoid this difficulty, we utilized extended (real *32) precision to precompute thecoefficients D0 – D5 for values of kh at appropriately chosen nodes on the boundary of thesquare Ω = [0, 1]×[0, 1] in the complex plane, and used Lagrange interpolation to evaluate D0
– D5 for arbitrary points in Ω to 13 digits (see [8] for a detailed description of the technique).Thus, in all of our numerical experiments reported in Section 5 below, the coefficients D0 –D5 were obtained by interpolation, rather than computed “from scratch”.
Now, we are ready to formulate Theorem 3.10, which is the principal analytical tool ofthis paper (together with Lemmas 6.6, 6.7, 6.8). Theorem 3.10 describes the 10th-ordercenter-corrected quadrature formula for the Green’s function for the Helmholtz equation intwo dimensions; this theorem is a special case of the high-order center-corrected trapezoidalrule for singular functions in two dimensions (see Theorem 3.4) with p = 3, and s(x, y) =
H(1)0 (k
√
x2 + y2). The 4th-order, 6th-order, and 8th-order center-corrected quadratures aresimilar and listed in Appendix B (see Lemma 6.6, 6.7 and 6.8). All the proofs are quitesimilar to that of Theorem 3.4, and are omitted.
Theorem 3.10 Suppose that n ≥ 1 is an integer, and a, h are two positive real numberssuch that h = a/n. Suppose further that φ : R2 → C is a function such that φ ∈ c8(R× R),and that φ is zero outside the square [−a, a] × [−a, a]. Then, for any k ∈ C
+,∫ a
−a
∫ a
−a
φ(x, y) ·H0(k√
x2 + y2) dx dy = Uτh(φ ·H0) +O(h10). (100)
In (100),
Uτh(φ ·H0) = T ′2D(φ ·H0) +
∑
p,q∈S
τhpqφ(ph, qh), (101)
whereS = p, q ∈ Z : |p+ q| ≤ 3 and |p− q| ≤ 3, (102)
T ′2D(φ ·H0) =
∑
(p,q)6=(0,0)
(
φ(ph, qh) ·H(1)0 (k
√
(ph)2 + (qh)2))
· h2, (103)
and
τh00 = D0 −
49
18
D1
h2+
7
9
D2
h4+
3
2
D3
h4− 1
18
D4
h6− 1
2
D5
h6, (104)
τh±10 = τh
0±1 =3
4
D1
h2− 13
48
D2
h4− 19
24
D3
h4+
1
48
D4
h6+
7
24
D5
h6, (105)
τh±20 = τh
0±2 = − 3
40
D1
h2+
1
12
D2
h4+
1
24
D3
h4− 1
120
D4
h6− 1
24
D5
h6, (106)
τh±30 = τh
0±3 =1
180
D1
h2− 1
144
D2
h4+
1
720
D4
h6, (107)
τh±1±1 =
5
12
D3
h4− 1
6
D5
h6, (108)
19
Figure 1: The 25 correction nodes
τh±1±2 = τh
±2±1 = − 1
48
D3
h4+
1
48
D5
h6. (109)
Remark 3.5 For simplicity, we assume here the function φ to be zero outside the square[−a, a]×[−a, a]. Thus, the endpoint corrected trapezoidal rule T ′2D,n
β2m+1(φ·H0) in Theorem 3.4
reduces to the standard trapezoidal rule T ′2D(φ · H0), and the integral and the sum on thesquare [−a, a] × [−a, a] are identical to those in R2. This simplification allows the directuse of the analytical formulae for D0 – D5 (see Lemma 3.9 above and Lemmas 6.1 – 6.5 inAppendix A).
Remark 3.6 Combining Remark 3.3 with the definitions (104)–(109), we observe that eachof the coefficients τh
pq in (104)–(109) has the form h2 · f(k ·h); we will refer to the coefficientsτhpq as correction coefficients.
Remark 3.7 The set S defined in (102) contains 25 pairs of integers (p, q); in other words,corrections at 25 points around the singularity are required to construct a 10th-order quadra-ture formula (see Figure 1). In general, for any integer p ≥ 0, 2p2 + 2p+ 1 correction nodesare needed to obtain a quadrature of order 2p+ 4.
3.4 Fast Numerical Application of Discretized Lippmann-Schwinger
Operators
In this section, we combine the 10th-order quadrature formula for the integral (100) with theFFT to obtain a fast procedure for the application of discretizations of the operator (12).We will denote by D the square [−a, a] × [−a, a] in R2.
Suppose that N ≥ 1 is an integer, h is a positive real number, and S is a set definedin (102). Suppose further that the coefficients τh
i1i2are defined in (104)–(109). Then, we
define a two-dimensional complex sequence H = Hi1i2, i1, i2 = −2N, ..., 2N, as follows:
Hi1i2 = H0(k√
(i1h)2 + (i2h)2) + τhi1i2/h2, (110)
20
when (i1, i2) ∈ S and (i1, i2) 6= (0, 0);
H00 = τh00/h
2, (111)
andHi1i2 = H0(k
√
(i1h)2 + (i2h)2), (112)
otherwise. We define a complex sequence Φ = Φi1i2, i1, i2 = −N, ..., N, to be the two-dimensional sequence defined by the formula
Φi1i2 = φ(i1h, i2h), (113)
where φ : R2 → C is a two-dimensional c8-function which is zero outside D.
Lemma 3.11 Suppose that the integers n, l1, l2 are such that n ≥ 1, −n ≤ l1 ≤ n, −n ≤ l2 ≤n, and that the real numbers a, x, y are such that a > 0, −a ≤ x ≤ a, −a ≤ y ≤ a. Supposefurther that φ : R2 → C is a c8-function which is zero outside the square [−a, a] × [−a, a].Then for any k ∈ C
+,∫ a
−a
∫ a
−a
φ(x′, y′) ·H(1)0 (k
√
(x− x′)2 + (y − y′)2) dx′ dy′
=∑
−n≤i1≤n
∑
−n≤i2≤n
Φi1i2 ·H(l1−i1)(l2−i2) +O(h10), (114)
where h = a/n, x = l1h, y = l2h, the two-dimensional sequence Φ = Φi1i2, i1, i2 = −n, ..., nis defined in (113), and the two-dimensional sequence H = Hj1j2, j1, j2 = −2n, ..., 2n isdefined in (110) – (112).
Proof. Due to (100), (101),∫ a
−a
φ(x′, y′) ·H(1)0 (k
√
(x− x′)2 + (y − y′)2) dx′ dy′
=∑
(i1,i2)∈I′
φ(i1h, i2h) ·H(1)0 (k
√
(l1h− i1h)2 + (l2h− i2h)2) · h2
+∑
(p,q)∈S
τhpq φ(l1h+ ph, l2h + qh) +O(h10), (115)
whereI ′ = i1, i2 ∈ Z : |i1| ≤ n, |i2| ≤ n, (i1, i2) 6= (l1, l2), (116)
and S is defined in (102). Now, (114) follows immediately from the combination of (115)and the definitions in (110) – (113).
Remark 3.8 Obviously, (114) is the Toeplitz convolution of the two-dimensional sequencesΦ, H , and as such, it can be rapidly calculated via the FFT (see Section 2.3 above). Thus,
∑
−n≤i1≤n
∑
−n≤i2≤n
Φi1i2 ·H(l1−i1)(l2−i2) = (F2n)−1(F2n(Φ′) · F2n(H)
)
l1l2, (117)
21
where −n ≤ l1 ≤ n, −n ≤ l2 ≤ n, and the two-dimensional sequence Φ′ = Φ′ij, i, j =
−2n, ..., 2n, is defined by
Φ′ij =
Φij if |i| ≤ n and |j| ≤ n0 if |i| > n or |j| > n
. (118)
Remark 3.9 For any point x outside the square [−a, a] × [−a, a], integral (12) is approx-imated via the standard trapezoidal rule. This approximation is 10th-order convergent, aslong as φ ∈ c10(R2) .
4 Description of the Procedure
This section describes the algorithm of the present paper in some detail. We start with aninformal description, follow with a more detailed one, and finish with a complexity analysis.
4.1 Informal Description of the Algorithm
Below, we describe an FFT-based 10th-order iterative algorithm for the solution of theLippmann-Schwinger equation
ψ(x) − k2
∫
D
Gk(x, y)V (y)ψ(y) dy = k2
∫
D
Gk(x, y)V (y)φ(y) dy (119)
in two dimensions, where D = [−a, a]× [−a, a], Gk is the Green’s function for the Helmholtz
equation in two dimension, i.e., Gk(x, y) = − i4· H(1)
0 (k‖x − y‖), and V (x) denotes thepotential at a point x. Here, ψ(x) and φ(x) are the scattered and the incident fields at apoint x, respectively.
As discussed in Remark 3.9, once the scattered field ψ in the domain D is known, thescattered field ψ outside D can be calculated via the standard trapezoidal rule applied to(119). Therefore, we focus on obtaining the solution of (119) for x ∈ D. Obviously, (119)can be written as the linear system
(1− A)ψ = Aφ, (120)
where ψ is the unknown scattered field in D, φ is the given incident field in D, 1 is theidentity operator, and A is the integral operator in (119). As discussed in Section 3, weuse (114) to approximate the integral operator A on the functions ψ, φ. With the help ofFFT (see Remark 3.8), we apply the discretized version of A rapidly to arbitrary vectors, andsolve the linear system (120) iteratively. We use one of the most popular iterative solvers,BI-CGSTAB (the stabilized bi-conjugate gradient method) (see [10], [12]).
4.2 Detailed Description of the Algorithm
Comment [Choose principal parameters.]
22
Set the size of the scattering structure to [−a, a] × [−a, a].Set the initial position of a point source to (x0, y0) to generate the incident field.Choose precision ǫ to be achieved for the iterative solver.Choose an integer n; set h = a
n; set the number of nodes discretizing a side of the square
to 2n+ 1, so that the total number of nodes in the discretization is N = (2n+ 1)2.Choose the wave number k for the incident and the scattered fields.Construct a two-dimensional sequence Vij, i, j = −n, ..., n via the formula Vij =
V (ih, jh).
Stage 1
Comment [Construct the values of the the Green’s function.]
For the user-specified h and k, calculate the correction coefficients D0, D1, D2, D3, D4, D5
in (71)–(76) via interpolation (see Remark 3.4).Construct the two-dimensional sequence H via the formulae (110) – (112) on the square
[−2a, 2a] × [−2a, 2a], and calculate its Fourier transform using the two-dimensional FFT.
Stage 2
Comment [Construct the right hand side of the linear system (120).]
For a point source (x0, y0), construct a two-dimensional sequence Φ = Φij, i, j =−n, ..., n for the discretized incident field on the domain [−a, a]×[−a, a] via the formula (113).Construct the two-dimensional sequence f = Φij · Vij, i, j = −n, ..., n.
As in Remark 3.8, use the two-dimensional FFT to calculate the Toeplitz convolution ofthe sequences H and f .
Stage 3
Comment [Solve the linear system using iterative solvers.]
Use the iterative solver BI-CGSTAB to solve the linear system (1 − A)ψ = Aφ to thepredetermined precision ǫ. The multiplication Aψ is done via the combination of FFT andthe Toeplitz convolution of the two-dimensional sequences H and g, where g = Ψij · Vij,i, j = −n, · · · , n with Ψij = ψ(ih, jh) (see Remark 3.8).
The solution is the scattered field at the N discretization points in the square [−a, a] ×[−a, a].
Stage 4
Comment [Calculate the scattered field at any point in the two-dimensional plane.]
Use interpolation to obtain the scattered field at any arbitrary point in the square [−a, a]×[−a, a], based on the scattered field at the N discretization points. As in Remark 3.9, applythe trapezoidal rule to (119) to obtain the scattered field at any arbitrary point outside thesquare [−a, a] × [−a, a].
23
4.3 Complexity Analysis
A brief analysis of the complexity of the algorithm is given below.In stage 1, the construction of the two-dimensional sequence H costs O(N), where N is
the total number of the discretization points on the square [−a, a]×[−a, a], i.e., N = (2n+1)2.The two-dimensional FFT costs O(N log(N)). Thus, the CPU time cost of the stage 1 is ofthe order O(N log(N)).
In stage 2, the construction of the two dimensional sequences Φ, f costs O(N), and thetwo-dimensional FFT costs O(N log(N)). Thus, the CPU time cost of the stage 2 is of orderO(N log(N)).
The CPU time cost of the stage 3 is of order O(Niter · N log(N)), where Niter is thenumber of iterations required by the iterative solver to get the pre-determined precision ǫ.
In stage 4, the CPU time cost of interpolating the field at any point in [−a, a] × [−a, a]O(N).
Summing up the CPU times above, we obtain the time estimate for the algorithm
T = α(Niter ·N log(N)) + β ·N + γ, (121)
where N is the total number of discretization points, Niter is the number of iterations requiredby the iterative solvers to reach the precision ǫ, and the coefficients α, β, γ are determinedby the computer system, implementation, etc.
The storage requirements of the algorithm are determined by the total number of dis-cretization points N and the number of iterations K performed before restarting the iterativesolvers, and are of the form
S = O(K ·N). (122)
5 Numerical Examples
The algorithm of Section 4 has been implemented in FORTRAN 77 in double precision.In this section, we illustrate the performance of the scheme as applied to two scatteringobjects: a Gaussian and a crude model of the human skull. The experiments were carriedout on a 2.8 GHz Pentium D desktop with 2 Gb of RAM and an L2 cache of 1 Mb. Thecalculations reported in Tables 1 and 3 were carried out with a requested accuracy of 10−13;the calculations reported in Tables 2 and 4 were carried out with a requested accuracy of10−9. We restarted the BI-CGSTAB every 5 steps.
Tables 1 - 4 illustrate the numerical behavior of the scattered field at arbitrary far-fieldpoints, generated by the potential V defined in (3); the incident field is produced by a singlepoint source. In Tables 1 and 2, we set the potential V (x, y) = e−40(x2+y2). Tables 3 and 4illustrate the numerical behavior of the scattered field, generated by a model of the humanskull, as shown in Figures 2 and 3. The headings of the Tables are as follows:
k is the wave number defined in (2);the computational grid is N ×N for a total of N2 discretization points;the computational grid is sizeobj wavelengths × sizeobj wavelengths;Nλ is the number of discretization points per wavelength;
24
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8
-0.6-0.4
-0.2 0
0.2 0.4
0.6 0.8
1
-0.1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Figure 2: The human skull model
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8
-0.6-0.4
-0.2 0
0.2 0.4
0.6 0.8
1
-0.1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Figure 3: The human skull model viewed from the top
Table 1: 10th order convergence of the algorithm for Gaussian objectsk N sizeobj Nλ Erel Niter tCPU
25 50 8λ 6.28 6.33E-06 16 1.2E-0125 100 8λ 12.6 6.63E-09 16 5.9E-0125 200 8λ 25.1 6.04E-12 16 2.6E+0025 400 8λ 50.2 7.25E-13 16 1.1E+0125 800 8λ 100 6.32E-13 16 5.5E+0125 1600 8λ 201 - 16 2.4E+02
25
Table 2: Gaussian objects with a fixed number of points per wavelengthk N sizeobj Nλ Erel Niter tCPU
25 50 8λ 6.28 6.33E-06 14 1.1E-0150 100 16λ 6.28 3.80E-06 20 7.2E-01100 200 32λ 6.28 4.44E-06 33 5.2E+00200 400 64λ 6.28 8.26E-06 61 4.4E+01400 800 128λ 6.28 1.60E-05 171 6.2E+02800 1600 255λ 6.28 - 891 1.4E+04
Table 3: 10th order convergence of the algorithm for the simulated human skullk N sizeobj Nλ Erel Niter tCPU
25 50 8λ 6.28 1.18E-04 134 1.1E+0025 100 8λ 12.6 1.91E-07 133 4.7E+0025 200 8λ 25.1 2.05E-10 134 2.1E+0125 400 8λ 50.2 4.56E-12 135 9.7E+0125 800 8λ 100 7.55E-12 132 4.7E+0225 1600 8λ 201 - 132 2.0E+03
Table 4: The simulated human skull with a fixed number of points per wavelengthk N sizeobj Nλ Erel Niter tCPU
25 50 8λ 6.28 1.17E-04 97 7.6E-0150 100 16λ 6.28 1.55E-05 165 5.8E+00100 200 32λ 6.28 1.03E-05 328 5.2E+01200 400 64λ 6.28 1.69E-05 756 5.5E+02400 800 128λ 6.28 2.21E-05 3286 1.2E+04800 1600 255λ 6.28 - 13568 2.1E+05
26
Erel is the average of the relative errors of the solution for the scattered field at twentychosen far-field points;
Niter is the number of iterations used by the BI-CGSTAB;tCPU is the CPU time required in seconds.The following observations can be made from the tables above, and from the more detailed
numerical tests performed by the authors.1. For smooth scattering objects, the numerical algorithm of Section 4 displays 10th-
order convergence; the CPU time required to obtain requested precision is proportional toNiter ·N2 logN , where N2 is the total number of discretization points, Niter is determined bythe requested precision, the number of iterations before restarting the iterative solver, thesize and the structure of the scattering objects.
2. For sufficiently smooth scatterers, the relative precision of the solution is determined bythe number of discretization points per wavelength. For example, to obtain 5-digit precision,we need roughly 6.5 points per wavelength. Thus, with our constraint of 2 GB RAM, fivedigits can be obtained for scattering objects as large as 300 wavelengths × 300 wavelengths.
3. The number of iterations increases dramatically as the size of the scattering objectincreases, as shown in Tables 2, 4.
6 Conclusions
In this paper, we construct an iterative algorithm for the solution of two-dimensional forwardscattering problems. The scheme is based on the combination of high-order quadratureformulae, rapid numerical application of the integral operator in the Lippmann-Schwingerequation, and the stabilized bi-conjugate gradient method (BI-CGSTAB). As illustrated viaseveral numerical examples, the scheme is (2p+ 4)th (p = 0, 1, 2, 3, ...) order convergent; thecomputational complexity of the algorithm is O(Niter ·N2 logN), where Niter is the numberof iterations used by the iterative solver, and N2 is the total number of discretization points.
The approach we use for the design of high order center-corrected quadrature formulaeintroduced in this paper is not limited to functions of the form (13) in two dimensions; itis also applicable to functions of the form (14) in three dimensions, as well as many similarsituations. Furthermore, the method does not require access to each of the functions P , Qseparately in (13) and (14); it only requires the evaluation of the whole kernel K given a pairof points (x, y). Quadrature formulae of order higher than 10 can also be constructed, thoughthe derivations become more tedious. Finally, the scheme is easily extended to rectangularregions of the form [−a, a] × [−b, b], even though this paper only discusses on the squareregion [−a, a] × [−a, a].
Acknowledgments. The authors would like to thank Mark Tygert for helpful discus-sions.
27
Appendix A
Lemmas 6.1 – 6.5 below provide analytical formulae for
∫ ∞
−∞
∫ ∞
−∞
(
xi−1yj−1H0(k√
x2 + y2))
dx dy−∑
(p,q)6=(0,0)
(
(ph)i−1(qh)j−1H0(k√
(ph)2 + (qh)2))
·h2,
(123)with (i, j) = (3, 1), (5, 1), (3, 3), (7, 1), (5, 3). The proofs are straightforward and tedious,and use the help of Mathematica.
Lemma 6.1 For any k ∈ C+ and h > 0,
D1 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr)x2 dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · (ph)2 · h2
=4
π
∫ ∞
−∞
dλ√k2 − λ2
·(
2
(iα1)3(iα2)− h4
2· e
iα1h(eiα1h + 1)(eiα2h + 1)
(eiα1h − 1)3(eiα2h − 1)
)
(124)
=4
π
∫ ∞
−∞
dλ√k2 − λ2
· 2
α31α2(1 + x1)3(1 + x2)
·(
(α2h)2
12+ 3z1 + z2 + 3y2
1 +3
2y1(iα1h) + 3y1y2 +
3
2y1(iα2h) +
3
2y2(iα1h)
+x31 + 3x2
1x2 + x31x2 −
1
2iα2hy2 +
1
2α2
1h2(x2
1 + 2x1)
+3
4α1α2h
2(x1x2 + x1 + x2) +1
4iα2
1α2h3(1 + x1)
2(1 + x2)
)
, (125)
where r, α1, α2, x1, x2, y1, y2, z1, z2 are defined by (88), (89), and (93)–(95).
28
Lemma 6.2 For any k ∈ C+ and h > 0,
D2 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr) x4 dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · (ph)4 · h2
=4
π
∫ ∞
−∞
dλ√k2 − λ2
·(
24
(iα1)5(iα2)− h6
2· e
iα1h + 11e2iα1h + 11e3iα1h + e4iα1h
(eiα1h − 1)5· e
iα2h + 1
eiα2h − 1
)
=4
π
∫ ∞
−∞
dλ√k2 − λ2
· 1
α51α2(1 + x1)5(1 + x2)
·(
2(iα2h)2 + (iα2h)
3 + 5(iα1h)(iα2h)2 +
3
10(iα2h)
4 +20
3(iα1h)
2(iα2h)2
+5
2(iα1h)(iα2h)
3 +1
2(iα1h)
4(iα2h) + 4x1(iα1h)4 +
45
2x1(iα1h)
3(iα2h)
+2x1(iα1h)4(iα2h) + 45x2
1(iα1h)3 + 25x2
1(iα1h)2(iα2h) + 6x2
1(iα1h)4
+45
2x2
1(iα1h)3(iα2h) + 3x2
1(iα1h)4(iα2h) + 15x3
1(iα1h)3 + 4x3
1(iα1h)4
+15
2x3
1(iα1h)3(iα2h) + 2x3
1(iα1h)4(iα2h) + x4
1(iα1h)4 +
1
2x4
1(iα1h)4(iα2h)
+15
2x2(iα1h)
3(iα2h) +1
2x2(iα1h)
4(iα2h) + 50x1x2(iα1h)2(iα2h)
+45
2x1x2(iα1h)
3(iα2h) + 2x1x2(iα1h)4(iα2h) + 25x2
1x2(iα1h)2(iα2h)
+45
2x2
1x2(iα1h)3(iα2h) + 3x2
1x2(iα1h)4(iα2h) +
15
2x3
1x2(iα1h)3(iα2h)
+2x31x2(iα1h)
4(iα2h) +1
2x4
1x2(iα1h)4(iα2h) + 12w2(iα2h) + 30z2(iα1h)(iα2h)
+40y2(iα1h)2(iα2h) + 15y1(iα1h)(iα2h)
2 + 30y1y2(iα1h)(iα2h) − 120u1 − 24u2
−180w1(iα1h) − 60w2(iα1h) − 60w1(iα2h) − 160z1(iα1h)2 − 90z1(iα1h)(iα2h)
−80z2(iα1h)2 + 35y1(iα1h)
3 − 40y1(iα1h)2(iα2h) − 130y2
1(iα1h)2
−180y21(iα1h)(iα2h) − 240y3
1(iα1h) − 120y31(iα2h) − 120y4
1 − 30y2(iα1h)3
−180y1y2(iα1h)2 − 360y2
1y2(iα1h) − 240y31y2 − 20z1(iα2h)
2 − 240z21 − 120z1z2
−360y21(iα1h) − 240y3
1 − 120y21(iα2h) − 240y1y2(iα1h) − 240y2
1y2
−24x51 − 120x4
1x2 − 24x51x2
)
, (126)
where
w1 = z1 −(iα1h)
3
24=
∞∑
n=4
(iα1h)n
(n+ 1)!, w2 = z2 −
(iα2h)3
24=
∞∑
n=4
(iα2h)n
(n+ 1)!, (127)
u1 = w1 −(iα1h)
4
120=
∞∑
n=5
(iα1h)n
(n+ 1)!, u2 = w2 −
(iα2h)4
120=
∞∑
n=5
(iα2h)n
(n + 1)!, (128)
and r, α1, α2, x1, x2, y1, y2, z1, z2 are defined by (88), (89) and (93)–(95).
29
Lemma 6.3 For any k ∈ C+ and h > 0,
D3 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr) x2 y2 dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · (ph)2(qh)2 · h2
=4
π
∫ ∞
−∞
dλ√k2 − λ2
(
4
(iα1)3(iα2)3− h6 · e
iα1h(eiα1h + 1)
(eiα1h − 1)3· e
iα1h(eiα1h + 1)
(eiα1h − 1)3
)
(129)
=4
π
∫ ∞
−∞
dλ√k2 − λ2
· −4
α31α
32(1 + x1)3(1 + x2)3
·(
1
240(iα1h)
4 +1
240(iα2h)
4 +3
4z1(iα1h)
2 +3
4z2(iα2h)
2
+9
2(iα1h)(iα2h)(z1 + z2) +
15
4z1(iα2h)
2 +15
4z2(iα1h)
2 − 3
2y1(iα1h)
2(iα2h)
−3
2y2(iα1h)(iα2h)
2 + 3u1 + 3u2 +9
2w2(iα1h) +
9
2w1(iα2h)
+3
2w1(iα1h) +
3
2w2(iα2h) −
9
4(iα1h)(iα2h)(z1 + z2) + 3z2
1 + 3z22
+9z1z2 −1
2y1(iα1h)
3 − 1
2y2(iα2h)
3 − 1
2y2
1(iα1h)2 − 1
2y2
2(iα2h)2
+9
4(iα1h)(iα2h)(3y1y2 +
1
6y1(iα2h) +
1
6y2(iα1h)) +
9
8y1(iα1h)
2(iα2h)
+3
8y1(iα2h)
3 +9
4y2
1(iα1h)(iα2h) +9
4y2
1(iα2h)2 +
3
2y3
1(iα2h)
+3
8y2(iα1h)
3 +9
8y2(iα1h)(iα2h)
2 +9
4y1y2(iα1h)
2 +9
4y1y2(iα2h)
2
+9
2y2
1y2(iα1h) + 9y21y2(iα2h) + 3y3
1y2 +9
4y2
2(iα1h)2 +
9
4y2
2(iα1h)(iα2h)
+9y1y22(iα1h) +
9
2y1y
22(iα2h) + 9y2
1y22 +
3
2y3
2(iα1h) + 3y1y32
−1
4(iα1h)
2(iα2h)2(2x1 + x2
1 + 2x2 + 4x1x2 + 2x21x2 + x2
2
+2x1x22 + x2
1x22) +
3
2y2
1(iα1h) + y31 +
3
2y2
2(iα2h) + y32 +
9
2y2
1(iα2h)
+9y1y2(iα1h + iα2h) + 9y21y2 +
9
2y2
2(iα1h) + 9y1y22
−3
4(iα1h)
2(iα2h)(x21 + 2x1x2 + x2
1x2)
−3
4(iα1h)(iα2h)
2(x22 + 2x1x2 + x1x
22) + 3x3
1x22 + 3x2
1x32 + x3
1x32
)
, (130)
where r, α1, α2, x1, x2, y1, y2, z1, z2, w1, w2, u1, u2 are defined by (88), (89), (93)–(95), (127)and (128).
30
Lemma 6.4 For any k ∈ C+ and h > 0,
D4 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr) x6 dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · (ph)6 · h2
=4
π
∫ ∞
−∞
dλ√k2 − λ2
·(
720
α71α2
− h8
2· e
iα1h + 57e2iα1h + 302e3iα1h + 302e4iα1h + 57e5iα1h + e6iα1h
(eiα1h − 1)7· e
iα2h + 1
eiα2h − 1
)
=4
π
∫ ∞
−∞
dλ√k2 − λ2
· 1
α71α2(1 + x1)7(1 + x2)
· (A+B), (131)
where
A = 60(α2h)2 − 30(iα2h)
3 − 210(iα1h)(iα2h)2 − 9(α2h)
4 − 105(α1h)(α2h)3
−385(α1h)2(α2h)
2 + 37800y21(iα1h) + 25200y3
1 + 7560y21(iα2h)
+15120y1y2(iα1h) + 15120y21y2 + 15120x5
1 + 5040x61 + 720x7
1 + 25200x41x2
+15120x51x2 + 5040x6
1x2 + 720x71x2 + 5040u1 + 720u2 + 12600w1(iα1h)
−360w2(iα2h) + 2520w2(iα1h) + 2520w1(iα2h) − 17220(α1h)2z1
+6300z1(iα1h)(iα2h) + 1260z2(α1h)(α2h) − 4620(α1h)2z2 + 15120z2
1
+2940y1(iα1h)3 − 6090y1(α1h)
2(iα2h) − 34440y21(α1h)
2 − 18900y21(α1h)(α2h)
+50400y31(iα1h) + 12600y3
1(iα2h) + 25200y41 + 3150y2(iα1h)
3
−18900y1y2(α1h)2 + 37800y2
1y2(iα1h) + 25200y31y2 − 840z1(α2h)
2 + 5040z1z2
+2310y2(α1h)2(iα2h) + 630y1(iα1h)(α2h)
2 + 1260y1y2(α1h)(α2h)
−63(iα1h)5 − 301(iα1h)
4(iα2h) + (α1h)6 +
63
2(α1h)
5(α2h) +1
2(α1h)
6(iα2h)
−2408x1(α1h)4 − 3150x1(α1h)
3(α2h) − 315x1(iα1h)5 − 1204x1(iα1h)
4(iα2h)
+6x1(α1h)6 +
315
2x1(α1h)
5(α2h) + 3x1(α1h)6(iα2h) − 6300x2
1(iα1h)3
+1680x21(α1h)
2(iα2h) − 3612x21(α1h)
4 − 3150x21(α1h)
3(α2h) − 630x21(iα1h)
5
−1806x21(iα1h)
4(iα2h) + 15x21(α1h)
6 + 315x21(α1h)
5(α2h) +15
2x2
1(α1h)6(iα2h)
−2100x31(iα1h)
3 − 2408x31(α1h)
4 − 1050x31(α1h)
3(α2h) − 630x31(iα1h)
5
−1204x31(iα1h)
4(iα2h) + 20x31(α1h)
6 + 315x31(α1h)
5(α2h) + 10x31(α1h)
6(iα2h)
−602x41(α1h)
4 − 315x41(iα1h)
5 − 301x41(iα1h)
4(iα2h) + 15x41(α1h)
6
+315
2x4
1(α1h)5(α2h) +
15
2x4
1(α1h)6(iα2h) − 63x5
1(iα1h)5 + 6x5
1(α1h)6
+63
2x5
1(α1h)5(α2h) + 3x5
1(α1h)6(iα2h) + x6
1(α1h)6 +
1
2x6
1(α1h)6(iα2h), (132)
31
B = −1050x2(α1h)3(α2h) − 301x2(α1h)
4(iα2h) +63
2x2(α1h)
5(α2h)
+1
2x2(α1h)
6(iα2h) + 3360x1x2(α1h)2(iα2h) − 3150x1x2(α1h)
3(α2h)
−1204x1x2(α1h)4(iα2h) +
315
2x1x2(α1h)
5(α2h) + 3x1x2(α1h)6(iα2h)
+1680x21x2(α1h)
2(iα2h) − 3150x21x2(α1h)
3(α2h) − 1806x21x2(α1h)
4(iα2h)
+315x21x2(α1h)
5(α2h) +15
2x2
1x2(α1h)6(iα2h) − 1050x3
1x2(α1h)3(α2h)
−1204x31x2(α1h)
4(iα2h) + 315x31x2(α1h)
5(α2h) + 10x31x2(α1h)
6(iα2h)
−301x41x2(α1h)
4(iα2h) +315
2x4
1x2(α1h)5(α2h) +
15
2x4
1x2(α1h)6(iα2h)
+63
2x5
1x2(α1h)5(α2h) + 3x5
1x2(α1h)6(iα2h) +
1
2x6
1x2(α1h)6(iα2h), (133)
and where r, α1, α2, x1, x2, y1, y2, z1, z2, w1, w2, u1, u2 are defined by (88), (89), (93)–(95),(127) and (128).
Lemma 6.5 For any k ∈ C+ and h > 0,
D5 =
∫ ∞
−∞
∫ ∞
−∞H
(1)0 (kr) x4 y2 dx dy −
∑
(p,q)6=(0,0)
H(1)0 (k
√
(ph)2 + (qh)2) · (ph)4(qh)2 · h2
=4
π
∫ ∞
−∞
dλ√k2 − λ2
·(
48
α51α
32
− h8 · eiα1h + 11e2iα1h + 11e3iα1h + e4iα1h
(eiα1h − 1)5· e
iα2h · (eiα2h + 1)
(eiα2h − 1)3
)
=4
π
∫ ∞
−∞
dλ√k2 − λ2
· 1
α51α
32(1 + x1)5(1 + x2)3
· (C +D) (134)
where
C =1
5(α2h)
4 + 240u1 + 360w1(iα1h) + 144u2 + 360w2(iα1h)
+360w1(iα2h) + 72w2(iα2h) − 320z1(α1h)2 − 540z1(α1h)(α2h)
−480z2(α1h)2 − 300z1(α2h)
2 − 180z2(α1h)(α2h) − 36z2(α2h)2 + 480z2
1
+720z1z2 + 144z22 − 70y1(iα1h)
3 − 240y1(α1h)2(iα2h) − 210y1(iα1h)(α2h)
2
+30y1(iα2h)3 − 260y2
1(α1h)2 − 1080y2
1(α1h)(α2h) − 360y21(α2h)
2
+480y31(iα1h) + 720y3
1(iα2h) + 240y41 + 180y2(iα1h)
3 + 120y2(iα1h)2(iα2h)
+30y2(iα1h)(α2h)2 − 1080y1y2(α1h)
2 − 1260y1y2(α1h)(α2h)
−180y1y2(α2h)2 + 2160y2
1y2(iα1h) + 1440y21y2(iα2h) + 1440y3
1y2 − 360y22(α1h)
2
−180y22(α1h)(α2h) + 1440y1y
22(iα1h) + 360y1y
22(iα2h) + 1440y2
1y22 + 120y3
2(iα1h)
+240y1y32 + 720y2
1(iα1h) + 480y31 + 720y2
1(iα2h) + 1440y1y2(iα1h)
+1440y21y2 + 720y1y2(iα2h) + 360y2
2(iα1h) + 720y1y22 + 72y2
2(iα2h) + 48y32 + 48x5
1
+720x41x2 + 144x5
1x2 + 1440x31x
22 + 720x4
1x22 + 144x5
1x22 + 480x2
1x32 + 480x3
1x32, (135)
32
D = 240x41x
32 + 48x5
1x32 − 24y2(iα2h)
3 + 24y22(α2h)
2 − 3(iα1h)4(iα2h)
−15(iα1h)3(iα2h)
2 + (α1h)4(α2h)
2 − 8x1(α1h)4 − 135x1(α1h)
3(α2h)
−100x1(α1h)2(α2h)
2 − 12x1(α1h)4(iα2h) − 45x1(iα1h)
3(iα2h)2 + 4x1(α1h)
4(α2h)2
−90x21(iα1h)
3 − 150x21(iα1h)
2(iα2h) − 12x21(α1h)
4 − 135x21(α1h)
3(α2h)
−50x21(α1h)
2(α2h)2 − 18x2
1(α1h)4(iα2h) − 45x2
1(iα1h)3(iα2h)
2 + 6x21(α1h)
4(α2h)2
−30x31(iα1h)
3 − 8x31(α1h)
4 − 45x31(α1h)
3(α2h) − 12x31(α1h)
4(iα2h)
−15x31(iα1h)
3(iα2h)2 + 4x3
1(α1h)4(α2h)
2 − 2x41(α1h)
4 − 3x41(α1h)
4(iα2h)
+x41(α1h)
4(α2h)2 − 45x2(α1h)
3(α2h) − 100x2(α1h)2(α2h)
2 − 3x2(α1h)4(iα2h)
−30x2(iα1h)3(iα2h)
2 + 2x2(α1h)4(α2h)
2 + 300x1x2(α1h)2(iα2h)
+120x1x2(iα1h)(α2h)2 − 135x1x2(α1h)
3(α2h) − 200x1x2(α1h)2(α2h)
2
−12x1x2(α1h)4(iα2h) − 90x1x2(iα1h)
3(iα2h)2 + 8x1x2(α1h)
4(α2h)2
+150x21x2(α1h)
2(iα2h) − 135x21x2(α1h)
3(α2h) − 100x21x2(α1h)
2(α2h)2
−18x21x2(α1h)
4(iα2h) − 90x21x2(iα1h)
3(iα2h)2 + 12x2
1x2(α1h)4(α2h)
2
−45x31x2(α1h)
3(α2h) − 12x31x2(α1h)
4(iα2h) − 30x31x2(iα1h)
3(iα2h)2
+8x31x2(α1h)
4(α2h)2 − 3x4
1x2(α1h)4(iα2h) + 2x4
1x2(α1h)4(α2h)
2
+60x22(iα1h)(α2h)
2 − 50x22(α1h)
2(α2h)2 − 15x2
2(iα1h)3(iα2h)
2 + x22(α1h)
4(α2h)2
+60x1x22(iα1h)(α2h)
2 − 100x1x22(α1h)
2(α2h)2 − 45x1x
22(iα1h)
3(iα2h)2
+4x1x22(α1h)
4(α2h)2 − 50x2
1x22(α1h)
2(α2h)2 − 45x2
1x22(iα1h)
3(iα2h)2
+6x21x
22(α1h)
4(α2h)2 − 15x3
1x22(iα1h)
3(iα2h)2 + 4x3
1x22(α1h)
4(α2h)2
+x41x
22(α1h)
4(α2h)2, (136)
where r, α1, α2, x1, x2, y1, y2, z1, z2, w1, w2, u1, u2 are defined by (88), (89), (93)–(95), (127)and (128).
Appendix B
Here, we present the center-corrected quadrature formulae of orders 4, 6 and 8 for the integral∫ a
−a
∫ a
−a
φ(x, y) ·H0(k√
x2 + y2) dx dy. (137)
Lemma 6.6 (The 4th-Order Center-corrected Quadrature Formula). Suppose thatn ≥ 1 is an integer, and a, h are two positive real numbers such that h = a/n. Supposefurther that φ : R2 → C is a function such that φ ∈ c2(R × R), and that φ is zero outsidethe square [−a, a] × [−a, a]. Then, for any k ∈ C
+,∫ a
−a
∫ a
−a
φ(x, y) ·H0(k√
x2 + y2) dx dy = Uτh
(
φ(x, y) ·H0(k√
x2 + y2))
+O(h4). (138)
In (138),
Uτh
(
φ(x, y) ·H0(k√
x2 + y2))
= T ′h(
φ(x, y) ·H0(k√
x2 + y2))
+∑
p,q∈S
τhpqφ(ph, qh), (139)
33
whereS = p, q ∈ Z : p = 0 and q = 0, (140)
T ′h(
φ(x, y) ·H0(k√
x2 + y2))
=∑
(p,q)6=(0,0)
(
φ(ph, qh) ·H0(k√
(ph)2 + (qh)2))
· h2, (141)
andτh00 = D0. (142)
Lemma 6.7 ( The 6th-Order Center-corrected Quadrature Formula). Suppose thatn ≥ 1 is an integer, and a, h are two positive real numbers such that h = a/n. Supposefurther that φ : R2 → C is a function such that φ ∈ c4(R × R), and that φ is zero outsidethe square [−a, a] × [−a, a]. Then, for any k ∈ C
+,
∫ a
−a
∫ a
−a
φ(x, y) ·H0(k√
x2 + y2) dx dy = Uτh
(
φ(x, y) ·H0(k√
x2 + y2))
+O(h6). (143)
In (143),
Uτh
(
φ(x, y) ·H0(k√
x2 + y2))
= T ′h(
φ(x, y) ·H0(k√
x2 + y2))
+∑
p,q∈S
τhpqφ(ph, qh), (144)
whereS = p, q ∈ Z : |p+ q| ≤ 1 and |p− q| ≤ 1, (145)
T ′h(
φ(x, y) ·H0(k√
x2 + y2))
=∑
(p,q)6=(0,0)
(
φ(ph, qh) ·H0(k√
(ph)2 + (qh)2))
· h2, (146)
and
τh00 = D0 − 2
D1
h2, (147)
τh±10 = τh
0±1 =1
2
D1
h2. (148)
Lemma 6.8 ( The 8th-Order Center-corrected Quadrature Formula). Suppose thatn ≥ 1 is an integer, and a, h are two positive real numbers such that h = a/n. Supposefurther that φ : R2 → C is a function such that φ ∈ c6(R × R), and that φ is zero outsidethe square [−a, a] × [−a, a]. Then, for any k ∈ C
+,
∫ a
−a
∫ a
−a
φ(x, y) ·H0(k√
x2 + y2) dx dy = Uτh
(
φ(x, y) ·H0(k√
x2 + y2))
+O(h8). (149)
In (149),
Uτh
(
φ(x, y) ·H0(k√
x2 + y2))
= T ′h(
φ(x, y) ·H0(k√
x2 + y2))
+∑
p,q∈S
τhpqφ(ph, qh), (150)
whereS = p, q ∈ Z : |p+ q| ≤ 2 and |p− q| ≤ 2, (151)
34
T ′h(
φ(x, y) ·H0(k√
x2 + y2))
=∑
(p,q)6=(0,0)
(
φ(ph, qh) ·H0(k√
(ph)2 + (qh)2))
· h2, (152)
and
τh00 = D0 −
5
2
D1
h2+
1
2
D2
h4+D3
h4, (153)
τh±10 = τh
0±1 =2
3
D1
h2− 1
6
D2
h4− 1
2
D3
h4, (154)
τh±20 = τh
0±2 = − 1
24
D1
h2+
1
24
D2
h4, (155)
τh±1±1 =
1
4
D3
h4. (156)
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